Modeling Rates and Proportions in SAS – 7
6. SIMPLEX MODEL
Dispersion models proposed by Jorgensen (1997) can be considered a more general case of Generalized Linear Models by McCullagh and Nelder (1989) and include a dispersion parameter describing the distributional shape. The simplex model developed by Barndorff-Nielsen and Jorgensen (1991) is a special dispersion model and is useful to model proportional outcomes. A simplex model has the density function given by
F(Y) = (2 * pi * sigma ^ 2 * (Y * (1 – Y)) ^ 3) ^ (-0.5) * EXP((-1 / (2 * sigma ^ 2)) * d(Y; Mu))
where d(Y; Mu) = (Y – Mu) ^ 2 / (Y * (1 – Y) * Mu ^ 2 * (1 – Mu) ^ 2) is a unit deviance function.
Similar to the Beta model, a simplex model also consists of 2 components. The first is a sub-model to describe the expected mean Mu. Since 0 < Mu < 1, the logit link function can be used to specify the relationship between the expected mean and covariates X such that LOG(Mu / (1 – Mu)) = X`B. The second is a sub-model to describe the pattern of dispersion parameter sigma ^ 2 also by a set of covariates Z such that LOG(sigma ^ 2) = Z`G. Due to the similar nature of parameterization between Beta model and Simplex model, model performances of these 2 often have been compared with each other. However, it is still an open question which model is able to outperform its competitor.
Similar to the case of Beta model, there is no out-of-box procedure in SAS estimating the simplex model. However, following its density function, we are able to model the simplex model with NLMIXED procedure as given below.